Open problems in the geometry and analysis of banach spaces

Antonio J. Guirao · Vicente Montesinos Václav Zizler Open Problems in the Geometry and Analysis of Banach Spaces Open Problems in the Geometry and Analysis of Banach Spaces Antonio J Guirao • Vicente Montesinos Václav Zizler Open Problems in the Geometry and Analysis of Banach Spaces 123 Antonio J Guirao Departamento de Matemática Aplicada Instituto de Matemática Pura y Aplicada Universitat Politècnica de València Valencia, Spain Vicente Montesinos Departamento de Matemática Aplicada Instituto de Matemática Pura y Aplicada Universitat Politècnica de València Valencia, Spain Václav Zizler Department of Mathematical and Statistical Sciences University of Alberta Alberta, Canada ISBN 978-3-319-33571-1 DOI 10.1007/978-3-319-33572-8 ISBN 978-3-319-33572-8 (eBook) Library of Congress Control Number: 2016939618 © Springer International Publishing Switzerland 2016 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland Dedicated to the memory of Joram Lindenstrauss, Aleksander Pełczy´nski, and Manuel Valdivia Preface This is a commented collection of some easily formulated open problems in the geometry and analysis on Banach spaces, focusing on basic linear geometry, convexity, approximation, optimization, differentiability, renormings, weak compact generating, Schauder bases and biorthogonal systems, fixed points, topology, and nonlinear geometry The collection consists of some commented questions that, to our best knowledge, are open In some cases, we associate the problem with the person we first learned it from We apologize if it may turn out that this person was not the original source If we took the problem from a recent book, instead of referring to the author of the problem, we sometimes refer to that bibliographic source We apologize that some problems might have already been solved Some of the problems are longstanding open problems, some are recent, some are more important, and some are only “local” problems Some would require new ideas, and some may go only with a subtle combination of known facts All of them document the need for further research in this area The list has of course been influenced by our limited knowledge of such a large field The text bears no intentions to be systematic or exhaustive In fact, big parts of important areas are missing: for example, many results in local theory of spaces (i.e., structures of finite-dimensional subspaces), more results in Haar measures and their relatives, etc With each problem, we tried to provide some information where more on the particular problem can be found We hope that the list may help in showing borders of the present knowledge in some parts of Banach space theory and thus be of some assistance in preparing MSc and PhD theses in this area We are sure that the readers will have no difficulty to consider as well problems related to the ones presented here We believe that this survey can especially help researchers that are outside the centers of Banach space theory We have tried to choose such open problems that may attract readers’ attention to areas surrounding them Summing up, the main purpose of this work is to help in convincing young researchers in functional analysis that the theory of Banach spaces is a fertile field of research, full of interesting open problems Inside the Banach space area, the text should help a young researcher to choose his/her favorite part to work in This vii viii Preface way we hope that problems around the ones listed below may help in motivating research in these areas For plenty of open problems, we refer also to, e.g., [AlKal06, BenLin00, BorVan10, CasGon97, DeGoZi93, Fa97, FHHMZ11, FMZ06, HaJo14, HMZ12, HMVZ08, HajZiz06, Kal08, LinTza77, MOTV09, Piet09, Woj91], and [Ziz03] To assist the reader, we provided two indices and a comprehensive table referring to the listed problems by subject We follow the basic notation in the Banach space theory and assume that the reader is familiar with the very basic concepts and results in Banach spaces (see, e.g., [AlKal06, Di84, FHHMZ11, LinTza77, HMVZ08, Megg98]) For a very basic introduction to Banach space theory—“undergraduate style”—we refer to, e.g., [MZZ15, Chap 11] By a Banach space we usually mean an infinite-dimensional Banach space over the real field—otherwise we shall spell out that we deal with the finite-dimensional case If no confusion may arise, the word space will refer to a Banach space Unless stated otherwise, by a subspace we shall mean a closed subspace The term operator refers to a bounded linear operator, and an operator with real values will be called a functional, understanding, except if it is explicitly mentioned, that it is continuous A subspace Y of a Banach space X is said to be complemented if it is the range of a bounded linear projection on X The unit sphere of the Banach space X, fx X W kxk D 1g, is denoted by SX , and the unit ball fx X W kxk Ä 1g is denoted by BX The words “smooth” and “differentiable” have the same meaning here Unless stated otherwise, they are meant in the Fréchet (i.e., total differential) sense If they are meant in the Gâteaux (i.e., directional) sense, we clearly mention it (for those concepts, see their definitions in, e.g., [FHHMZ11, p 331]) We say that a norm is smooth when it is smooth at all nonzero points Sometimes, we say that “a Banach space X admits a norm k k,” meaning that it admits an equivalent norm k k By ZFC we mean, as usual, the ZermeloFraenkel-Choice standard axioms of set theory Unless stated otherwise, we use this set of axioms We say that some statement is consistent if its negation cannot be proved by the sole ZFC set of axioms Cardinal numbers are usually denoted by @, while ordinal numbers are denoted by ˛, ˇ, etc With the symbol @0 we denote the cardinal number of the set N of natural numbers, and @1 is the first uncountable cardinal Similarly, !0 (sometimes denoted !) is the ordinal number of the set N under its natural ordering, and !1 is the first uncountable ordinal The continuum hypothesis then reads 2@0 D @1 The cardinality 2@0 of the set of real numbers (the continuum) will be denoted by c If no confusion may arise, we sometimes denote by !1 also its cardinal number @1 We prepared this little book as a working companion for [FHHMZ11] and [HMVZ08] We often use this book to upgrade and update information provided in these two references Overall, we would be glad if this text helped in providing a picture of the present state of the art in this part of Banach space theory We hope that the text may serve also as a kind of reference book for this area of research Preface ix Acknowledgments The third named author would like to express his gratitude to the late Joram Lindenstrauss and Olek Pełczy´nski for their lifelong moral support and encouragement to the Prague Banach space group, in particular in connection with the organization of Prague annual international winter schools for young researchers In fact, for the young, starting Prague group, the moral support of the Israeli and Polish schools in the seventies and eighties of the last century was vital The second named author is grateful for Olek’s encouraging attitude regarding mathematics and his personal friendship and the first and second named authors to the Prague group for its continued support, encouragement, and friendship All three authors want to dedicate this work also to the founder of the modern Spanish functional analysis school, the late Manuel Valdivia The authors thank the Springer team, especially Marc Strauss, for their interest in this text and Mr Saswat, Mishra, for his professional editing of the manuscript The authors thank their colleagues who helped by various means, advice or references, etc., to this text The third named author appreciates the electronic access to the library of the University of Alberta The first two named authors want also to thank the Universitat Politècnica de València, its Instituto de Matemática Pura y Aplicada, and its Departamento de Matemática Aplicada, for their support and the working conditions provided The authors were also supported in part by grants MTM2011-22417 and MICINN and FEDER (Project MTM2014-57838-C2-2-P) (Vicente Montesinos) and Fundación Séneca (Project 19368/PI/14), and MICINN and FEDER (Project MTM2014-57838-C2-1-P) (Antonio J Guirao) The material comes from the interaction with many colleagues in meetings, in work, and in private conversations and, as the reader may appreciate in the comments to the problems, from many printed sources—papers, books, reviews, and even beamers from presentations—and, last, from our own research work It is clear then that it will be impossible to explicitly thank so many influences The authors prefer to carry on their own shoulders the responsibility for the selection of problems, eventual inaccuracies, wrong attributions, or lack of information about solutions The names of authors appearing in problems, in comments, and in the reference list correspond to the panoply of mathematicians to whom thanks and acknowledgment usually appear in the introduction to a book Above all, the authors are indebted to their families for their moral support and encouragement The authors wish the readers a pleasant time spent over this little book Valencia, Spain Valencia, Spain Calgary, Canada 2016 A.J Guirao V Montesinos V Zizler .. .Open Problems in the Geometry and Analysis of Banach Spaces Antonio J Guirao • Vicente Montesinos Václav Zizler Open Problems in the Geometry and Analysis of Banach Spaces 123 Antonio... help in convincing young researchers in functional analysis that the theory of Banach spaces is a fertile field of research, full of interesting open problems Inside the Banach space area, the. .. denoted !) is the ordinal number of the set N under its natural ordering, and !1 is the first uncountable ordinal The continuum hypothesis then reads 2@0 D @1 The cardinality 2@0 of the set of real